Rationalise the denominator of $\frac{\sqrt{7}}{\sqrt{2}}$
Give your answer in its simplest form.
The objective is to simplify the expression \(\frac{\sqrt{7}}{\sqrt{2}}\) by rationalizing the denominator. Rationalizing the denominator means rewriting the fraction so that the denominator contains no square roots or irrational numbers. This is a standard procedure in algebra that helps in simplifying expressions for easier comparison or further calculation.
To rationalize the denominator of a fraction like \(\frac{\sqrt{7}}{\sqrt{2}}\), we multiply both the numerator and the denominator by the square root present in the denominator. In this case, that value is \(\sqrt{2}\). Multiplying the fraction by \(\frac{\sqrt{2}}{\sqrt{2}}\) is mathematically equivalent to multiplying by 1, so the value of the original expression remains unchanged. This results in the following expression:
\[\frac{\sqrt{7}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{7} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}\]
Now we simplify the numerator and the denominator. Using the property of square roots that \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\), the numerator becomes:
\[\sqrt{7} \times \sqrt{2} = \sqrt{7 \times 2} = \sqrt{14}\]
For the denominator, we know that \(\sqrt{2} \times \sqrt{2} = 2\). Combining these results, we get:
\[\frac{\sqrt{14}}{2}\]
This expression is now in its simplest form, as the denominator is an integer.
Rationalizing denominators is frequently used in fields like physics and engineering. For example, when calculating the magnitude of a force vector or complex impedance in electrical circuits, results often involve square roots. Standardizing these results by removing radicals from the denominator makes it easier to compare different values, perform further arithmetic, or input results into specialized software programs that work best with rational denominators. This skill ensures that your mathematical expressions remain clean and easy to interpret.